I'm referring this paper here and I was trying to work out the calculation of the energy density which is the 00th element of the energy momentum tensor $T_{\mu\nu}$ which is given as: $$ T_{\mu \nu}=\partial_\mu \Phi^* \partial_\nu \Phi+\partial_\nu \Phi^* \partial_\mu \Phi-\eta_{\mu \nu}\left(\eta^{\alpha \beta} \partial_\alpha \Phi^* \partial_\beta \Phi-m^2 \Phi^* \Phi\right) $$ here, the metric tensor $\eta_{\mu\nu}$ is given by: $$ \eta_{\mu\nu}=diag(1,-1,-r^2, -r^2sin^2\theta) $$ I know that under the static spherically symmetric conditions, only the 00th element of the energy momentum tensor is non zero which is the energy density. In order to calculate it, I put $\mu=\nu=0$ along with $\alpha=\beta=0$. Upon calculating it, I got: $$ T_{00}= m^2|\Phi|^2 $$ however, in the paper the energy density comes out to twice of what I got i.e. $T_{00}=2m^2|\phi|^2$. What am I doing wrong? Is this not how I am supposed to calculate the energy density in this particular case? Any help in this regard would be greatly appreciated.
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$\begingroup$ What am I doing wrong? You are forgetting the contribution of spatial gradient of scalar field to the energy density. $\endgroup$– A.V.S.Commented Oct 5, 2023 at 13:00
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$\begingroup$ Why did you set $\alpha=\beta=0$? These are contracted indices that you sum over. $\endgroup$– GhosterCommented Oct 6, 2023 at 0:34
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